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The Zen of Magic Squares, Circles, and Stars


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Table of Contents

Preface xi Acknowledgments xix Introduction 1 CHAPTER ONE: Magic Construction 37 CHAPTER TWO: Classification 65 CHAPTER THREE: Gallery I : Squares, Cubes, and Tesseracts 147 CHAPTER FOUR: Gallery 2: Circles and Spheres 297 CHAPTER FIVE: Gallery 3: Stars, Hexagons, and Other Beauties 325 Some Final Thoughts 369 Notes 375 For Further Reading 395 Index 397 About the Author 403

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A refreshing new look at a timeless topic, brimming over with ideas, littered with surprising twists. Anyone who loves numbers, anyone who enjoys puzzles, will find The Zen of Magic Squares, Circles, and Stars compulsive (and compulsory!) reading. -- Ian Stewart, University of Warwick, author of "Flatterland" and "Does God Play Dice?" Pickcover carries the mystique of magic squares and their relatives into the twenty-first century with his new book, The Zen of Magic Squares, Circles, and Stars. Whether you're seeking an introduction to magic squares, an in-depth study, some historical information, or just some enjoyable magic figure problems, tricks, properties, or novelties-this book is for you. Pickover does not leave a magic square unturned, and tantalizes us just enough to want to explore further. From the very famous to the less known, Pickover brings them all together in this amazing collection as he points out their roles in science, our lives and the universe. -- Theoni Pappas author of "The Joy of Mathematics" and "Math-A-Day" At first glance magic squares may seem frivolous (Ben Franklin's opinion, even as he spent countless hours studying them!), but I think that is wrong. The great nineteenth-century German mathematician Leopold Kronecker said 'God Himself made the whole numbers--everything else is the work of men,' and Cliff Pickover's stimulating book hints strongly at the possibility that God may have done more with the integers than just create them. I don't believe in magic in the physical world, but magic squares come as close as we will probably ever see to being mathematical magic. -- Paul J. Nahin, University of New Hampshire, author of "Duelling Idiots and Other Probability Puzzlers" Clifford Pickover has compiled such a wonderfully voluminous collection of magic squares and related configurations that the physical book itself threatens to take the shape of a magic cube. Whether or not you achieve arithmetical satori contemplating these engagingly intricate patterns, you will surely come to appreciate their history, beauty, and richness. -- John Allen Paulos, Temple University, author of "Innumeracy" and "A Mathematician Reads the Newspaper" Who would have thought that the simple numbers we used when learning to count as children could be arranged into so many geometric patterns with interesting properties? If playing with numbers is your thing, this latest work by Cliff Pickover will provide you will countless hours of mystical entertainment and mental challenges. -- Julien Clinton Sprott, University of Wisconsin-Madison Every generation seems to demand its own updated book dedicated to magic squares. Pickover's work meets the needs of the present generation. It is an enthusiastic, whimsical, and idiosyncratic compendium of magic squares and their variants, some of which are centuries old, some as fresh as corn picked this morning. -- Sherman Stein, University of California, Davis Well documented and illustrated, this book is essentially a huge catalogue of these arithmetical beasts called magic squares. No book that I know of presents such a large range of 'mysterious, odd, and fun' patterns, as the author puts it. This includes the numerous geometrical diagrams that reveal some hidden symmetries not obvious to the eye contemplating the number arrays. Throughout, Pickover convincingly substantiates his claim that the field of magic square study is still wide open. -- Arturo Sangalli, author of "The Importance of Being Fuzzy"

About the Author

Clifford Pickover is the author of over twenty books on a broad range of topics in science and art, a columnist for Odyssey, and an inventor. His books include Surfing Through Hyperspace: Understanding Higher Universes in Six Easy Lessons. Wonders of Numbers: Adventures in Mathematics. Mind, and Meaning, and The Loom of God: Mathematical Tapestries at the Edge of Time.


A perpetual idea machine, Clifford Pickover is one of the most creative, original thinkers in the world today. Journal of Recreational Mathematics Pickover just seems to exist in more dimensions than the rest of us. -- Ian Stewart Scientific American Clifford Pickover is many things--scientist, scholar, author, editor, and visionary... Games It is a safe bet to conjecture that this is the best recreational mathematics book that will be published in this year... Pickover writes with his usual style and straightforward simplicity in this book. The material is presented well and can be understood by anyone with a basic middle school mathematics background. This is a cool book! -- Charles Ashbacker Journal of Recreational Mathematics Through accessible and readable prose and through detailed, highquality line illustrations, Pickover ably transports the general reader from culturally embedded traditional topics to a new and surprising frontier. -- Harold Don Allen Mathematics Teacher Pickover writes about his subject with contagious enthusiasm and comprehensive erudition. Choice A splendid recreational book... An extremely alluring page-turner. -- Andrew Bremner Notices of the American Mathematical Society

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